# matrix proving problem

• Aug 4th 2010, 08:12 AM
mikai
matrix proving problem
Suppose u and v are solutions to the linear system Ax=b. Show that if scalars α and β satisfy α+β=1, then αu+βv is also a solution to the linear system Ax=b.
• Aug 4th 2010, 10:29 AM
Failure
Quote:

Originally Posted by mikai
Suppose u and v are solutions to the linear system Ax=b. Show that if scalars α and β satisfy α+β=1, then αu+βv is also a solution to the linear system Ax=b.

Just plug $\alpha u+\beta v$ for $x$ into the left side of $Ax=b$ and use the linearity of applying $A$ to that vector to simplify everything to just $b$

$Ax = A(\alpha u+\beta v)=\alpha (Au)+\beta (Av)=\ldots = b$